The relation between the flux ratio and energy ratio in convectively mixed layers

Two alternative formulations of the problem of entrainment at the top of a convecting surface layer are in current use. One expresses the mixing in terms of the ratio of the buoyancy flux at the top of the convecting layer to that at the ground, and the other (which is the more fundamental) compares the energy used for entrainment with that produced by the destabilizing flux at the ground. Several authors have proposed similar methods of calculating the latter and relating the two ratios, but we believe these authors are in error because of a fallacious assumption about the nature of the mixing in the practically uniform layer. The present note attempts to resolve this difficulty by using a simple model which considers explicitly the separate contributions of the environment and the surface flux to the density of the mixed layer. We suggest that the physically relevant energy ratio is the increase in potential energy due to the redistribution of the initial density profile compared with the potential energy made available by heating from below.